Article pubs.acs.org/JPCB
Mechanics of Nanostructured Porous Silica Aerogel Resulting from Molecular Dynamics Simulations Sandeep P. Patil,*,† Ameya Rege,‡ Sagardas,† Mikhail Itskov,‡ and Bernd Markert† †
Institute of General Mechanics, RWTH Aachen University, Templergraben 64, 52062 Aachen, Germany Department of Continuum Mechanics, RWTH Aachen University, Kackertstraße 9, 52072 Aachen, Germany
‡
S Supporting Information *
ABSTRACT: Silica aerogels are nanostructured, highly porous solids which have, compared to other soft materials, special mechanical properties, such as extremely low densities. In the present work, the mechanical properties of silica aerogels have been studied with molecular dynamics (MD) simulations. The aerogel model of 192 000 atoms was created with different densities by direct expansion of βcristobalite and subjected to series of thermal treatments. Because of the high number of atoms and improved modeling procedure, the proposed model was more stable and showed significant improvement in the smoothness of the resulting stress−strain curves in comparison to previous models. Resulting Poisson’s ratio values for silica aerogels lie between 0.18 and 0.21. The elasticity moduli display a power law dependence on the density, with the exponent estimated to be 3.25 ± 0.1. These results are in excellent agreement with reported experimental as well as computational values. Two different deformation scenarios have been discussed. Under tension, the low-density aerogels were more ductile while the denser ones behaved rather brittle. In the compression simulations of low-density aerogels, deformation occurred without significant increase in stress. However, for high densities, atoms offer a higher resistance to the deformation, resulting in a more stiff response and an early densification. The relationship between different mechanical parameters has been found in the cyclic loading simulations of silica aerogels with different densities. The residual strain grows linearly with the applied strain (≥0.16) and can be approximated by a phenomenological relation ϵp = 1.09ϵmax − 0.12. The dissipation energy also varies with the compressive strain according to a power law with an exponent of 2.31 ± 0.07. Moreover, the tangent modulus under cyclic loading varies exponentially with the compressive strain. The results of the study pave the way toward multiscale modeling of silica as well as reinforced silica aerogels.
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INTRODUCTION
The literature on the mechanical modeling of aerogels is not very exhaustive. Three prominent approaches undertaken to model aerogels are diffusion-limited cluster aggregation models,11−13 molecular dynamics simulation models,14−16 and micromechanical multiscale models.17,18 Coarse-grained models appeared to be useful to clarify certain linear-elastic properties of aerogels. These models are often subdivided further into two categories: hard-sphere aggregation models and flexible coarsegrained models. Aggregation models relevant to aerogels are diffusion-limited aggregation (DLA), reaction-limited aggregation (RLA), diffusion-limited cluster aggregation (DLCA), and reaction-limited cluster aggregation (RLCA).19 In DLA and RLA, clusters grow by addition of monomers, while in DLCA and RLCA clusters are themselves mobile and may meet and aggregate. Application of DLCA on silica aerogels was first reported by Hasmy et al.20,21 They compared the DLCA model of aerogels against short- and long-range results of the smallangle neutron scattering (SANS) spectra of colloidal aerogels. The results of DLCA were found to be in good agreement with
Silica aerogels belong to the largest class of highly porous inorganic aerogels. The literature on the synthesis and characterization of these aerogels has been expanding since their discovery in 1931 by Kistler.1,2 Silica aerogels are excellent candidates for a variety of applications, such as shock absorbers, drug carriers, Knudsen pumps, and in particular thermal superinsulations.3 For the latter, fields of applications are ranging from space technology, automotive, and process technology all the way to appliances and buildings.4 One major drawback in the applications so far is the brittle behavior, often resulting in crack formation or severe dust release. These effects are a consequence of the low tensile strength of silica aerogels, which results from their high porosity in combination with the low fraction of fully connected mass and the low intrinsic tensile strength of amorphous silica. The mechanical properties of silica aerogels have been investigated in detail using ultrasonic runtime analysis,5 mechanical testing,6 nanoindentation,7 in situ small-angle scattering,8 and inelastic scattering (Brillouin, Raman, neutron).9,10 Typically, Young’s modulus E is found to scale with the bulk density ρ as E ∝ ρ2.6−3.5. © 2017 American Chemical Society
Received: April 4, 2017 Revised: May 23, 2017 Published: May 23, 2017 5660
DOI: 10.1021/acs.jpcb.7b03184 J. Phys. Chem. B 2017, 121, 5660−5668
Article
The Journal of Physical Chemistry B
Figure 1. Schematic representation of the stages in the preparation process of silica aerogel: (a) β-cristobalite, (b) amorphous silica, and (c) silica aerogel of 0.62 g cm−3 density. Representative snapshots show silica aerogels of (d) 0.28, (e) 0.42, and (f) 1.18 g cm−3. For explanatory purposes, all the snapshots are shown in same size; however, the actual sizes of the simulation boxes are ∼28.363, ∼24.643, ∼21.743, and ∼17.523 nm3 for silica aerogel of 0.28, 0.42, 0.62, and 1.18 g cm−3 densities, respectively.
the SANS data. Later, Ma et al.11,12 modeled the aerogel structure using DLCA, where they described the interparticle bonds as beam elements and studied their linear elastic properties within the framework of finite element analysis. Of certain interest was the influence of dangling bonds to the mechanical response of aerogels. They concluded their reports by identifying the scaling exponent in the power-law relationship. Recently, the model by Ma et al.11,12 was extended to a two-level model by Liu et al.22 to simulate the tensile properties of silica aerogels. Specifically, Young’s modulus and the tensile strength were explored. MD simulations have gained significant attention toward aerogel modeling in the past decade. Gelb14 used flexible coarse-graining to model low-density silica aerogels and described their linear elastic properties, primarily the bulk modulus. In his first report, the value of the bulk modulus was found to be lower than the one obtained in experiments. In an extension study, Ferreiro-Rangel and Gelb15 further explored the bulk modulus using fluctuation analysis and direct compression−expansion simulations. The evaluated power-law exponent was between 3 and 3.15, which was then in good agreement with the reports by Ma et al.11,12 Recently, the uniaxial tensile-compressive response of silica aerogels of different densities predicted by MD simulations and hybrid Monte Carlo methods16 were reported. Young’s modulus obtained from both methods was in good agreement with the values in the literature, except for very low-density aerogels. The density was found to have a weak influence on Poisson’s ratio. Uniaxial tension and compression of large magnitude were reported. Low-density aerogels were found to behave more elastically in comparison to high density ones, which failed sooner, suggesting more brittle behavior. This is due to the fact that low-density aerogels have higher porosities, giving the atoms larger freedom to move under deformation. The models also displayed auxetic behavior at larger tensile strains. Despite such vast literature on molecular modeling of aerogels, the inelastic properties of silica aerogels under cyclic loading have so far not been investigated. In this case, many inelastic features like cyclic stress softening and residual
deformation can be observed. In this contribution, we attempt to investigate these effects in silica aerogels on the nanoscale via MD simulations. The effects of different model sizes (sample sizes) on the mechanical properties of these aerogels and their response under different strain rates are also studied which, to the best of our knowledge, surprisingly, has never been investigated before in a computational study. This paper is organized as follows. The Methods section discusses the methods applied to study the mechanical deformation of silica aerogels. In the Results and Discussion section, results of MD simulations of silica aerogels under tension and compression are analyzed. Special attention is focused on the study of damage mechanisms under cyclic loading.
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METHODS
Molecular dynamics (MD) simulations were carried out on silica aerogels using the program Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS).23 Mechanical properties of silica aerogel were estimated by performing mechanical tests, such as uniaxial tension and compression. Silica aerogels are characterized by a highly complex structure, wherein the atoms are arranged randomly. For the same density models, the major dissimilarity appears in the arrangement of arrays of the interconnected gel particles. For this reason, four different silica aerogel models were independently created for every analysis. In MD simulations, the interatomic potential plays a crucial role. It includes bonding, nonbonding interactions with neighbors, and their extensions. In this work, the interatomic potential by Vashishta et al.24,25 was used, which takes into account the direct interaction between two atoms associated by the distance of separation. This potential also considers the energy associated with the bonding angle and the orientation of three atoms. Hence, by incorporating both two- and three-body interactions, it can exactly replicate silica. Mathematically, the interaction potential is given by24,25 5661
DOI: 10.1021/acs.jpcb.7b03184 J. Phys. Chem. B 2017, 121, 5660−5668
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The Journal of Physical Chemistry B Vij =
Hij rij
ηij
+
ZiZj rij
⎛ −rij ⎞ ⎛ −rij ⎞ Pij exp⎜ ⎟ ⎟ − 4 exp⎜ ⎝ r1s ⎠ rij ⎝ r4s ⎠
Vijk = Bjik f (rij , rik)(cos θjik − cos θjik̃ )2
Table 1. Description of the MD Simulation Models crystalline box size (nm3)
(1) (2)
size size size size
where rij is the atomic distance between ith and jth atom and Vij and Vijk denote the two- and three-body potentials, respectively. The first term on the right-hand side of eq 1 represents the steric repulsion, where Hij and ηij are the strength and exponent of steric repulsion, respectively. Coulombic interactions are formulated in the second term, where Zi represents the charge of the ith atom. Pij represents the electric polarizability of the atom; r1s and r4s are the interaction cutoff distances. In eq 2, Bjik is the strength of the three body interaction, and θjik denotes the angle between the vectors rij and rik. A detailed description and the exact parameter values for these equations are given by Vashishta et al.24,25 Creating Numerical Model. The atomic model of silica aerogel was created starting with β-cristobalite atomic coordinates as seen in Figure 1a. The lattice is as given by Wyckoff.26 β-Cristobalite was taken as the starting sample because it has a density of 2.17 g cm−3 close to that of amorphous silica. Amorphous silica was computationally prepared from the crystalline silica by an annealing process. Using the LAMMPS package, periodic boundary conditions were assigned in three mutually perpendicular directions, i.e., bulk silica. The time step size of 0.5 fs was used for the velocityVerlet algorithm in order to solve the equations of motion of the particles. Random velocity was assigned to atoms at 7000 K. Subsequently, the sample was quenched to 300 K using NVT ensemble, which was followed by the system energy minimization using the conjugate gradient method. Finally, the sample was relaxed to atmospheric conditions (300 K and 1 bar) to generate amorphous silica (Figure 1b). The amorphous silica sample was relaxed at 300 K for 7.5 ps followed by instantaneous expansion to the desired density. The expanded sample was then heated to 3000 K for 50 ps followed by relaxation. Finally, the sample was quenched to 0 K followed by the energy minimization. The porous silica aerogel was formed when the sample is further brought back to the atmospheric conditions (Figure 1c). All the heat treatment operations were carried out using the NVT ensemble except for the relaxation which was performed using NPT ensemble. The change in the silica structure during the process of generation of the aerogel of 0.62 g cm−3 density is shown in Figures 1a−c. The range of the densities considered in this study was from 0.2 to 2.2 g cm−3. Figures 1d−f illustrate the typical porous silica aerogel of 0.28, 0.42, and 1.18 g cm−3 densities, respectively. Geometric Parameters. The radial distribution function (RDF), g(r), is calculated in LAMMPS. The value of r corresponding to the maximum value of g(r) represents the bond length between the atoms. For Si−O, Si−Si, and O−O the bond distances were calculated as 1.609, 3.066, and 2.626 Å, respectively. A detailed representation of the RDF is included in the Supporting Information. These parameters were well within the range provided by Murillo et al.27 and even validate the experimental results for bond length parameters for silica:24 1.61 ± 0.05 Å for Si−O, 3.08 ± 0.10 Å for Si−Si, and 2.632 ± 0.089 Å for O−O. In order to validate the size insensitivity of the silica aerogel models, different crystalline simulation boxes were considered (see Table 1).
0.28 0.33 0.42 0.62 1.18 1.58 2.08
1 2 3 4
aerogel box size (nm3)
size insensitivity 7.02 11.40 17.52 21.02 ρ values (g cm−3) in eq 4 14.32 28.36 14.32 26.70 14.32 24.64 14.32 21.73 14.32 17.52 14.32 15.91 14.32 14.66 5.73 9.31 14.32 17.18
number of atoms 12288 52728 192000 331776 192000 192000 192000 192000 192000 192000 192000
Mechanical properties of silica aerogel, primarily Young’s modulus and fracture strength, were estimated by simulating uniaxial tension. The deformation of the samples was done with a time step size of 0.5 fs and using NPT ensemble at the atmospheric conditions. Young’s modulus and fracture strength of the silica aerogel model with crystalline simulation box size of ∼14.323 nm3 were determined at different strain rates: 0.4, 0.04, 0.004, and 0.0004 ps−1. Along with uniaxial tension and compression simulations, compressive cyclic loading was also conducted on primarily three different density samples, 0.28, 0.42, and 0.62 g cm−3.
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RESULTS AND DISCUSSION Size and Rate Insensitivity. Although bulk silica models were considered, the models show size-sensitive mechanical properties. To avoid this size sensitivity of the silica aerogel models, uniaxial tensile simulations were carried out on different initial sizes (crystalline silica), with a density of 1.18 g cm−3 and the loading rate of 0.004 ps−1. Murillo et al.27 reported the crystalline sample size of about 7.13 nm3, which contained 24 000 atoms. A bulk crystal silica sample including 52 728 atoms was considered by Lei et al.28 Therefore, it is worth to discuss the significance of the simulation box size on the mechanical properties of the bulk crystal silica model. In this work, four bulk crystal silica samples were considered (see Table 1). In MD simulations, computational costs play a vital role; therefore, selection of the silica aerogel model is a balanced trade-off with numerical accuracy. Figure 2 shows the influence of the silica aerogel model size on the resultant mechanical properties. More specifically, the average values of the elasticity modulus and failure stress were studied. As the model size increases, the elasticity modulus and failure stress decrease significantly. Stability was observed when the sample size approaches 192 000 atoms (aerogel box size ∼17.523 nm3). Therefore, for the models of ∼17.523 and ∼21.023 nm3, there was nearly no significant difference in the elastic modulus. However, a very small variation in failure stress occurs due to the randomness associated with MD simulations. Moreover, it also suggests an improvement in the obtained results compared to the previous works.27,28 Much smoother stress−strain curves were obtained when the number of atoms in the simulation model was equal to or greater than 192 000 atoms (inset of Figure 2). Thus, all the simulations presented in this work were carried out for the silica aerogel model of 192 000 atoms (crystalline simulation box of ∼14.323 nm3), 5662
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from the measurement of longitudinal and transverse strains for silica aerogels of different densities, as shown in Figure 4. Our models exhibit Poisson’s ratios between 0.18 and 0.21, which is well within the experimental values.30−32
Figure 2. Results of MD simulations with standard error bars: Variation of the elastic modulus (square) and the failure stress (circle) with the size of the crystalline simulation box. Inset: stress−strain curves for the models of ∼5.733, ∼9.313, ∼14.323, and ∼17.183 nm3. Here, the crystalline simulation box sizes are the sizes of amorphous silica before the expansion into silica aerogel.
which was computationally more feasible than the 331 776 atoms model. For the strain rate sensitivity analysis, the model with a density of 1.18 g cm−3 was created and subjected to uniaxial tension under a NPT ensemble, up to the occurrence of complete failure. Figure 3 shows the resulting variation of the
Figure 4. Longitudinal versus transverse strain obtained from simulations of silica aerogels with density of 0.62 g cm−3. Inset: variation of Poisson’s ratio against the longitudinal strain with standard error bars.
Relationship between the Elastic Modulus and Density. Figure 5 shows the relationship obtained between
Figure 3. Results of MD simulations with standard error bars: influence of applied strain rate on the elastic modulus (square) and failure stress (circle).
elastic modulus and failure stress with the applied strain rate. An abrupt change in both the elastic modulus and the failure stress was observed when the strain rate decreases from 0.4 to 0.04 ps−1. For the strain rates 0.004 and 0.0004 ps−1, the computed mechanical properties were within the range of the standard deviation, which was also validated by Chowdhury et al.29 for amorphous silica. Thus, further MD simulations in this work were performed using a strain rate of 0.004 ps−1. Poisson’s Ratio. Poisson’s ratio of silica aerogels has so far been measured experimentally as well as computationally. Gross et al.30 reported values between 0.205 and 0.230, almost independent of the gel density. Gross and Scherer31 found it to be 0.2 in beam-bending experiments on a single specimen. Á lvarez-Arenas et al.32 also found it to be 0.2 from the analysis of ultrasonic resonances of air-surrounded aerogel plates. Moreover, Ferreiro-Rangel and Gelb16 reported values between 0.17 and 0.24 using simulations of a flexible coarse-grained model. In this computational study, it was calculated directly
Figure 5. Elasticity modulus versus density in log−log scale compared to previous MD simulations.27,28,33
the elasticity modulus and density, along with the ones reported by Murillo et al.27 and Lei et al.28,33 MD simulations were conducted for seven densities: 0.28 0.33, 0.42, 0.62, 1.18, 1.58, and 2.08 g cm−3. We observed an almost linear stress−strain response up to 0.04 of tensile strain. For this region, linear fits are plotted on the stress−strain curves in Figure 6. The corresponding Young’s modulus value for each density can be determined simply by E = σ/ϵ, and the obtained values are in good agreement with previous works.27,28,33 The resulting relationship between Young’s modulus and the density can be described by a power law. In our study, the exponential value for the relation was estimated to be 3.25 ± 0.1, and the complete expression can be given as 5663
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Table 2. Exponents for the Power Law Relation between Different Mechanical Parameters of Silica Aerogela relation
exponent in this work
exponent in literature
E and ρ
3.25 ± 0.1
3.2 ± 0.2b 3.3 ± 0.4c
σ and ρ
2.79 ± 0.1
2.3 ± 0.2b 2.53 ± 0.15c
range of densities ρ = 0.42−2.2 g cm−3; see ref 35 ρ = 0.23−2.2 g cm−3; see refs 27, 28 ρ = 0.42−2.2 g cm−3; see ref 35 ρ = 0.23−2.2 g cm−3; see ref 27
a
The data presented here were obtained for the average of three densities: 0.28, 0.42, and 0.62 g cm−3. bData obtained from experiments. cData obtained from simulations.
power law relation between the fracture strength σf (in MPa) and density ρ (kg m−3) can be expressed as σf = 4.3271 × 10−6 × ρ2.79
The exponent obtained was very close to the values from the literature (2.53 ± 0.1527 and 2.3 ± 0.235) (see Table 2). Tensile Stress−Strain Curve. We also studied the rupture strengths of the silica models under uniaxial tension. Figure 6a shows tensile stress−strain curves for different densities. A typical deformation evolution (rupture) process of a cubic specimen of 0.62 g cm−3 density is shown in Figure 6b. Three densities were considered, particularly, 0.28, 0.42, and 0.62 g cm−3. The corresponding porosities of the models before the start of deformation were calculated to be 0.872, 0.8037, and 0.7139, respectively. Under tensile loading, very less change in porosity was observed, which can be attributed to the lower bulk modulus of silica aerogel. Rupture of the model occurs at 0.25, 0.18, and 0.11 tensile strain for densities of 0.28, 0.42, and 0.62 g cm−3, respectively. For the density of 0.28 g cm−3, the constant strain rate of 0.004 ps−1 was applied for 7.1 ns, and a maximum strength of 44.2 MPa was observed. The corresponding strain was 0.25, and the porosity increased slightly to 0.8864 from the initial value of 0.872. Because of its low density, there exist enough voids for the atoms to take new positions. Hence, the breakage of atomic bonds happens in stages. Moreover, when the load was applied, clusters of atoms elongate initially due to high porosity, resulting in a high failure strain. For the density of 0.42 g cm−3, the load was applied for 3.3 ns. The maximum strength and strain were computed to 106.4 MPa and 0.18, respectively. For silica aerogels with the density of 0.62 g cm−3, mostly the maximum strength 253.7 MPa at a strain of 0.11 was obtained. Here, the failure pattern resembles the exact brittle fracture with sudden crack initiation and propagation within a very short time period of ∼20 ps. Beyond the maximal stress, decrease in stress happens in a gradual way due to the presence of unbroken bonds. Here, due to lesser porosity, atoms were more restricted and could not easily occupy new positions during the deformation process. Hence, breakage of bonds occurs all at the same moment leading to brittle fracture. Moreover, beyond the maximal strength point, stress decreases rapidly, which is typical for brittle fracture. Therefore, silica aerogel features the mechanical properties of a brittle material under the tensile loading. However, density plays an important role in the rupture strength, elongation at rupture and Young’s modulus. There was significant decrease of the tensile strength
Figure 6. (a) Representative tensile stress−strain curves from MD simulations, showing the behavior of 0.28, 0.42, and 0.62 g cm−3 density silica aerogels used for the calculation of mechanical properties. The determination of Young’s modulus via linear fits up to 0.04 of the tensile strain of densities 0.28 and 0.62 g cm−3 is illustrated. As an example, for density 0.28 g cm−3, let σ1 and ϵ1 denote the stress and strain for its linear regime. Then, Young’s modulus E0.28 is calculated as σ1/ϵ1. Similarly for density 0.62 g cm−3, E0.62 = σ2/ϵ2. (b) Snapshots at different strains during the tensile strain pulling simulations of 0.62 g cm−3 density silica aerogel. The undeformed simulation boxes are shown with dotted lines.
E = 1.4177 × 10−6 × ρ3.25
(4)
(3)
where E is the elasticity modulus of the model in MPa and ρ is density in kg m−3. In their experiments, Gross and Fricke34 found the value of the exponent to be 3.49 ± 0.07, while Woignier et al.35 reported the value of 3.2 ± 0.2 in the experiments on partially densified samples (0.42 and 2.2 g cm−3). Murillo et al.27 and Lei et al.28 obtained in computational simulations the exponent value as 3.11 ± 0.21 and 3.168, respectively. Thus, the proposed model of silica aerogel shows good agreement with previous studies, especially in comparison to the experimental work of by Woignier et al.35 (see Table 2). The range of densities modeled in this research is similar to the ones covered by Woignier et al.35 for partially densified samples. Similarly, the fracture strength of silica aerogel in dependence of the density has also been investigated (see detailed description in the Supporting Information). Accordingly, a 5664
DOI: 10.1021/acs.jpcb.7b03184 J. Phys. Chem. B 2017, 121, 5660−5668
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densities results in a higher resistance to this movement of atoms. This, in turn, leads to the early densification and the stiffer stress−strain response. Figure 7b shows the typical process of compressive deformation of silica aerogel with a density of 0.62 g cm−3. The rapid densification started at strains of approximately 0.65, 0.5, and 0.35 for different densities of 0.28, 0.42, and 0.62 g cm−3, respectively. Compressive Cyclic Loading. The cubic model of ∼21.733 nm3 of silica aerogel was loaded at different maximum strains of ∼0.10 to ∼0.70 and unloaded to zero stress as shown in Figure 8. The average stress−strain curve was obtained from
as well as Young’s modulus with the decreasing density, although the ductility seemed to increase. Young’s modulus was later calculated from the slope of the initial linear portion of the stress−strain plot by fitting a straight line. Compression Stress−Strain Curve. Figure 7a shows the stress−strain curves at three densities, 0.28, 0.42, and 0.62 g
Figure 8. Representative stress−strain curve of cyclic compressive loading of silica aerogel for ρ = 0.62 g cm−3. The red dotted lines show the initial and tangent moduli. The parameters σ0 and ϵ0 in the graph denote stress and strain in the linear regime, respectively, while Δσt and Δϵt represent increments of the stress and strain of the subsequent loading−unloading cycles, respectively. The numbers of loading− unloading cycles are indicated in blue circle.
six consecutive compressive loading and unloading cycles (ρ = 0.62 g cm−3). All stress−strain curves for loading−unloading cycles are included in Supporting Information for the densities 0.28 and 0.42 g cm−3 as well. Under cyclic compressive loading, one observes typical inelastic effects, such as residual deformation, hysteresis, and Mullins effect, which are known commonly from elastomers.40,41 In particular, with an increase of the applied strain amplitude, the cluster debonding increases, resulting in a total collapse of the fragmented structure. This results in more energy favorable configurations. Furthermore, negligible strain recovery is seen in the densification region. In the following, we study the influence of the density to the response of silica aerogels under cyclic compressive loading. Figure 9 shows the residual strain versus the maximal compressive strain. For higher strains (≥0.16), the residual strain grows linearly with the applied strain for all the considered densities. A straight line approximately follows the linear equation
Figure 7. (a) Stress−strain curves for silica aerogels of 0.28, 0.42, and 0.62 g cm−3 density under compression. (b) Snapshots of silica aerogel with 0.62 g cm−3 density at 0% (undeformed state), 24%, 40%, and 64% compression strain are applied. The undeformed simulation boxes are shown with dotted lines.
cm−3 computed under uniaxial compression. Here, the density has significant influence on the mechanical response of silica aerogels. These curves can be divided into three regions: (1) a linear elastic region (
